Mixing
Is Pull back of Lie group a Lie group?
$begingroup$
Suppose $f:Mrightarrow N$ is a submersion. Then, $Mtimes_NM$ is a smooth manifold from Transversal theorem.
Suppose $theta:Grightarrow H$ is a morphism of Lie groups. Assume that it is a submersion. Does it imply $Gtimes_H G$ is a Lie group?
As $Grightarrow H$ is submersion, $Gtimes_H G $ is a smooth manifold. Does it have to be Lie group?
When will $Gtimes_H G$ is a Lie group?
differential-geometry lie-groups smooth-manifolds
asked Jan 20 at 4:24
Praphulla KoushikPraphulla Koushik
17419
$endgroup$
add a comment |
$begingroup$
Suppose $f:Mrightarrow N$ is a submersion. Then, $Mtimes_NM$ is a smooth manifold from Transversal theorem.
Suppose $theta:Grightarrow H$ is a morphism of Lie groups. Assume that it is a submersion. Does it imply $Gtimes_H G$ is a Lie group?
As $Grightarrow H$ is submersion, $Gtimes_H G $ is a smooth manifold. Does it have to be Lie group?
When will $Gtimes_H G$ is a Lie group?
differential-geometry lie-groups smooth-manifolds
asked Jan 20 at 4:24
Praphulla KoushikPraphulla Koushik
17419
$endgroup$
add a comment |
$begingroup$
Suppose $f:Mrightarrow N$ is a submersion. Then, $Mtimes_NM$ is a smooth manifold from Transversal theorem.
Suppose $theta:Grightarrow H$ is a morphism of Lie groups. Assume that it is a submersion. Does it imply $Gtimes_H G$ is a Lie group?
As $Grightarrow H$ is submersion, $Gtimes_H G $ is a smooth manifold. Does it have to be Lie group?
When will $Gtimes_H G$ is a Lie group?
differential-geometry lie-groups smooth-manifolds
asked Jan 20 at 4:24
Praphulla KoushikPraphulla Koushik
17419
$endgroup$
Suppose $f:Mrightarrow N$ is a submersion. Then, $Mtimes_NM$ is a smooth manifold from Transversal theorem.
Suppose $theta:Grightarrow H$ is a morphism of Lie groups. Assume that it is a submersion. Does it imply $Gtimes_H G$ is a Lie group?
As $Grightarrow H$ is submersion, $Gtimes_H G $ is a smooth manifold. Does it have to be Lie group?
When will $Gtimes_H G$ is a Lie group?
differential-geometry lie-groups smooth-manifolds
differential-geometry lie-groups smooth-manifolds
asked Jan 20 at 4:24
Praphulla KoushikPraphulla Koushik
17419
asked Jan 20 at 4:24
Praphulla KoushikPraphulla Koushik
17419
asked Jan 20 at 4:24
Praphulla KoushikPraphulla Koushik
17419
asked Jan 20 at 4:24
Praphulla KoushikPraphulla Koushik
17419
asked Jan 20 at 4:24
Praphulla KoushikPraphulla Koushik
17419
17419
add a comment |
add a comment |
1 Answer
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$begingroup$
Yes it’s always a Lie group.
Closure under multiplication and inverses you can check, so it’s a group. For smoothness, note $Mtimes_NM$ isn’t just a manifold, but an embedded submanifold of $Mtimes M$. The multiplication and inversion for $Gtimes_HG$ are restricted from $Gtimes G$ to the submanifold, hence smooth.
answered Jan 20 at 4:57
BenBen
4,198617
$endgroup$
$begingroup$
Ok. Restriction of smooth map to embeded submanifold is smooth.. So, it i the case.. Thanks
$endgroup$
– Praphulla Koushik
Jan 20 at 5:35
$begingroup$
@PraphullaKoushik No problem!
$endgroup$
– Ben
Jan 20 at 6:00
$begingroup$
I have added more details as a community wiki answer. Let me know if I am missing something. Thank you :)
$endgroup$
– Praphulla Koushik
Jan 20 at 9:21
add a comment |
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1 Answer
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votes
1 Answer
1
active
oldest
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active
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votes
$begingroup$
Yes it’s always a Lie group.
Closure under multiplication and inverses you can check, so it’s a group. For smoothness, note $Mtimes_NM$ isn’t just a manifold, but an embedded submanifold of $Mtimes M$. The multiplication and inversion for $Gtimes_HG$ are restricted from $Gtimes G$ to the submanifold, hence smooth.
answered Jan 20 at 4:57
BenBen
4,198617
$endgroup$
$begingroup$
Ok. Restriction of smooth map to embeded submanifold is smooth.. So, it i the case.. Thanks
$endgroup$
– Praphulla Koushik
Jan 20 at 5:35
$begingroup$
@PraphullaKoushik No problem!
$endgroup$
– Ben
Jan 20 at 6:00
$begingroup$
I have added more details as a community wiki answer. Let me know if I am missing something. Thank you :)
$endgroup$
– Praphulla Koushik
Jan 20 at 9:21
add a comment |
$begingroup$
Yes it’s always a Lie group.
Closure under multiplication and inverses you can check, so it’s a group. For smoothness, note $Mtimes_NM$ isn’t just a manifold, but an embedded submanifold of $Mtimes M$. The multiplication and inversion for $Gtimes_HG$ are restricted from $Gtimes G$ to the submanifold, hence smooth.
answered Jan 20 at 4:57
BenBen
4,198617
$endgroup$
$begingroup$
Ok. Restriction of smooth map to embeded submanifold is smooth.. So, it i the case.. Thanks
$endgroup$
– Praphulla Koushik
Jan 20 at 5:35
$begingroup$
@PraphullaKoushik No problem!
$endgroup$
– Ben
Jan 20 at 6:00
$begingroup$
I have added more details as a community wiki answer. Let me know if I am missing something. Thank you :)
$endgroup$
– Praphulla Koushik
Jan 20 at 9:21
add a comment |
$begingroup$
Yes it’s always a Lie group.
Closure under multiplication and inverses you can check, so it’s a group. For smoothness, note $Mtimes_NM$ isn’t just a manifold, but an embedded submanifold of $Mtimes M$. The multiplication and inversion for $Gtimes_HG$ are restricted from $Gtimes G$ to the submanifold, hence smooth.
answered Jan 20 at 4:57
BenBen
4,198617
$endgroup$
Yes it’s always a Lie group.
Closure under multiplication and inverses you can check, so it’s a group. For smoothness, note $Mtimes_NM$ isn’t just a manifold, but an embedded submanifold of $Mtimes M$. The multiplication and inversion for $Gtimes_HG$ are restricted from $Gtimes G$ to the submanifold, hence smooth.
answered Jan 20 at 4:57
BenBen
4,198617
answered Jan 20 at 4:57
BenBen
4,198617
answered Jan 20 at 4:57
BenBen
4,198617
answered Jan 20 at 4:57
BenBen
4,198617
4,198617
$begingroup$
Ok. Restriction of smooth map to embeded submanifold is smooth.. So, it i the case.. Thanks
$endgroup$
– Praphulla Koushik
Jan 20 at 5:35
$begingroup$
@PraphullaKoushik No problem!
$endgroup$
– Ben
Jan 20 at 6:00
$begingroup$
I have added more details as a community wiki answer. Let me know if I am missing something. Thank you :)
$endgroup$
– Praphulla Koushik
Jan 20 at 9:21
add a comment |
$begingroup$
Ok. Restriction of smooth map to embeded submanifold is smooth.. So, it i the case.. Thanks
$endgroup$
– Praphulla Koushik
Jan 20 at 5:35
$begingroup$
@PraphullaKoushik No problem!
$endgroup$
– Ben
Jan 20 at 6:00
$begingroup$
I have added more details as a community wiki answer. Let me know if I am missing something. Thank you :)
$endgroup$
– Praphulla Koushik
Jan 20 at 9:21
$begingroup$
Ok. Restriction of smooth map to embeded submanifold is smooth.. So, it i the case.. Thanks
$endgroup$
– Praphulla Koushik
Jan 20 at 5:35
$begingroup$
Ok. Restriction of smooth map to embeded submanifold is smooth.. So, it i the case.. Thanks
$endgroup$
– Praphulla Koushik
Jan 20 at 5:35
$begingroup$
@PraphullaKoushik No problem!
$endgroup$
– Ben
Jan 20 at 6:00
$begingroup$
@PraphullaKoushik No problem!
$endgroup$
– Ben
Jan 20 at 6:00
$begingroup$
I have added more details as a community wiki answer. Let me know if I am missing something. Thank you :)
$endgroup$
– Praphulla Koushik
Jan 20 at 9:21
$begingroup$
I have added more details as a community wiki answer. Let me know if I am missing something. Thank you :)
$endgroup$
– Praphulla Koushik
Jan 20 at 9:21
add a comment |
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